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77m A nnals of P~biUty
1980, Vol. 8, No. 1, 83 95
VECTOR VALUED SUBADDITIVE PROCESSES
AND APPLICATIONS IN PROBABILITYt
BY N. GHoussouB AND J. MICHAEL STEELE
University of British Columbia and Stanford University
An ergodic theorem is proved which extends the subadditive ergodic theorem of Kingman and the Banach valued ergodic theorem of Mouner The theorem is applied to several problems, in particular to a problem on empirical distribution functions.
1. IntrMuction. The subadditive ergodic theorem of Kingman [2, 5, 6, 16, 17,
18, 211 is a nonfinear generalization of the Birkhoff ergodic theorem which, in the
ten years of its existence, has become established as one of the most useful results
in probability theory. Fifteen years before Kingman's result, Mourier [191 had
extended the Birkhoff theorem in an entirely different direction by proving that it
is valid in any Banach space. The main objective of this paper is to proye a result
which extends both the Kingman and Mourier theorems.
While the simple existence of such a generalization may be of interest, our primary motivation has been the feeling that such a result should be genuinely useful. To expedite possible applications we have outlined in the third section several general procedures which give rise to vector valued subadditivo processes. In particular we treat a problem there on empirical distribution functions which seems of independent interest.
A second motivation has been provided by the intimate relationship between the subadditive ergodic theorem and the geometry of the underlying Banach lattice. It is possible, for example, to characterize those spaces isomorphic to L' by means of subadditive processes.
Before yielding to the details of Section 2, it seems useful to give a less technical description of our main result. We suppose E is a Banach lattice (i.e., 1xl < lyl =* 11x11 %~ llyll), and we will call a sequence (S,) of Bochner integrable E valued functions a subadditive process provided for all natural n, k we have
(1. 1) Sn + k < S. + Sk 0 0 n
where 0 is a measure preserving transformation on the probability space (2, 9~, P). Our key result is that under a mild restriction on E, one has for any positive subadditive process that S.In converges in norm with probability one. In particular we are content to note here that E can be LP, 1 < p < oc, co, or essentially any familiar space except 1' or C(K).
Received July 14, 1978.
1 Supported by National Science Foundation Grant MCS77 16974.
AMS 1970 subject chusificatiOn. P~ 60FIS.
Key word, andphrases Ergodic theorem, subadditive ergodic theorem, Banach lattice, empirical distributions.
83
84 N. GHOUSSOUB AND J. MIC~ STEELE
2. Main results. We must first recall some facts from ergodic theory and functional analysis. When equality is assumed to hold in (1.1), (S,,) is called an additive process, and the key result about such processes is the following.
VECTOR VALUED EROODIC THEOREM. (Mourier [ 19]). If (S.) is an additive process valued in any Banach space E then S,,In converges a.s. in norm to E.'(Sj where I is the a field of 0 invariant sets. (Here and subsequently E' will denote the conditional expectation with respect to the a field 1).
REAL SUIBADDITIVE EROODIC THEOREM. (Kingman [16]). If (S.) is a real subadditiveprocess then inf n~E(S.) > oc implies
(2.1) litn S,,In a.s.
where
(2.2) limEI(S./n).
We need to recall some notions about Banach lattices which may not be familiar to probabilists. A Banach lattice E is called countably order complete (COC) provided for any nonempty countable B which is majorized by an element of E that sup B exists. Further, E has order continuous norm (OCN) provided E is COC and every decreasing positive sequence in E is norm convergent. Finally, E is weakly sequentially complete (WSC) if every increasing norm bounded sequence in E is norm convergent. (For background on these notions, one can consult Schaefer [21], pages 54 and 92.)
A sequence (x,,) in E will be called subadditive (superadditive) if X 1k X + Xk (XnIk > x,, + xk) for all natural n, k; we start by developing several elementary facts about subadditive sequences in Banach lattices. Finally, we let E' denote the dual of E and let E' denote the positive cone of E'.
LEMMA 1. In a Banach lattice E with an order continuous norm one has the following properties:
(2.3) If 0 .1~ x,, ,Q x and x. converges weakly to x
then x,, converges s~ to x.
(2.4) If 0 ~Q x,, < y,,, y,, converges s~ to x
and x,, converges weakly to the same x
then x,, converges ~ to x.
PROOF. This lemma has been established in Heinich [141, but we will provide an alternate proof. By the result of Diestel and Seifert 17] which sAys that a weakly compact order interval is the range of a countably additive vector measure, and the result of Anantharaman [8], page 270 which says the extreme points of such ranges are denting points we see that x is a denting point of [0, xl. Thus we have established the first part of the lemma. To prove the second half we note
SUBADDITIVE VECTOR PROCESSES 85
0 Q x,, < x,, v x < y,, v x hence for f E= E' we have f(x,,) < f(x,, V x) < f(y. V x) and thus x,, V x converges weakly to x. Since x, V x + x, A x = x, + x we also see x, A x converges weakly to x, which implies by (2.3) that x,, A x converges strongly to x. Since 0 4 x,, A x < x,, < y,, the strong convergence of the bounds on x,, implies (2.4).
PRoposmoN 1. For a countably order cowlete Banach lattice E the following conditions are equivalent:
(2.5) E has an order continuous norm.
(2.6) For every positive subadditive sequence (xn) in E,
the sequence x,,ln is norm convergent to inf(x./n).
PROOF. If (x,,) is subadditive, (x2k/2 k) is decreasing and assuming E has an order continuous norm this sequence is norm convergent to z infkx2k/2k = inf x.1n. Since applying any element of E' to x,, yields a real subadditive sequence we see that (xjn) is weak Cauchy. Since this sequence is valued in the weakly compact order interval [z, xl] it is weakly convergent to z. By Lemma 1 this is enough to imply norm convergence to z.
To prove that (2.6) implies (2.5) it suffices to note that if (y,,) is positive decreasing then (x,,) = (ny,,) is subadditive.
PRoposiTioN 2. For a Banach lattice E, the following conditions are equivalent:
(2.7) E is weakly sequentially conWIete.
(2.8) For every superadditive sequence (x,) in E such that sup j 1 xn J n < 00
the sequence (xjn) is norm convergent to sup x,,In.
PROOF. y,, = x,, xl is a positive superadditive sequence. Again, y2k12 k is
increasing and norm bounded, hence it is norm convergent to z = supy2k/2 k =
supjjn. y,,ln is weak Cauchy and valued in [0, z], hence it is weakly convergent
to z. Lemma 1 applies and gives the norm convergence.
We can now precisely state our main result.
THEOREm 1. For a countably order conplete Banach lattice E the following properties are equivalent:
(2.9) E has an order continuous norm.
(2.10) For every positive, E valued subadditive process (S,,) we have
strong convergence of (S,, In) with probability one.
86 N. GHOUSSOUB A" J. MIC~ STEELE
PROOF. Since (2.10) immediately implies (2.9) by Proposition 1, we focus on the more important implication of (2.10) by (2.9). If I denotes the invariant a field of 0, we first show
(2.11) V[ SnIn] converges strongly to Z = inf.El[SnInj for all £0.
Setting Y. VISJ we have Y. > 0 and
Y. k El[S.1k] ~Q El[Sn + Sk . On] = Yn + El[Sk 0"] = Y,, + Yk.
This proves for a.c. w, (Yn(w)) is a positive subadditive sequence in E so by Proposition 1 Y.In is norm convergent to Z = inf YnIn = inf E"[SnIn] a.e.
Now since (Sn) is a sequence of Bochner integrable functions it is almost separably valued, so there is no loss in assuming E is separable. For f E E E' we apply Kingman's theorem to the real process Xn = f(Sn) to obtain
(2.12) lim. 1 f(Sn) = lim E' 1 f(SJ) = inf V(If(SJ)
n ( n n
outside a set 2f with P(Cf) = 0. By (2.11), El(SnIn) converges. strongly to inf EI(SnIn) so by the continuity of f we have
S
lim E'(=) =f(inf El(SnIn)). n
Since bounded linear functionals commute with conditional expectation, by (2.12) and the preceeding equality we have for w iZ Of that
(2.13) hmf(S.1n) = lim E'( !J(S.)),
n
= Iiinj(EI(S./n) = f(inf El(S.1n)) = f(Z).
We now adapt a technique used by Kingman [181 page 195, to show that S.In is majorized by a norm convergent sequence of random variables. For that, we fix k > 0 and suppose n > k. Letting N = N(n) be the integral part of nlk, we see by subadditivity that
s,, 4 IN 1Sk . 0(r 1)k + S._Nk . ONk
(2.14) r_
4 2 Nr 1 Z1 + WN
Sk . 0(r l,k and W
N = Ijk IS. , ONk. Since WN has the same distribu
Where Z,. = _ 1 j
tion for all N and since Ell W, J1 < oo, we have for all e > 0 that
100_ lp(li WN11 > eN) = 200 IP(11 W, 11 > eN) 4 E 11 W, 11 < oo.
N N
The Borel Cantelli lemma thus implies that 11 WNIIIN converges a.e. to zero. Since IN
r Z,, is an additive process, we have by Mourier's theorem that a.e.
liM _L IN
(2.15) N r ,Z,. = EI[Z,] = El[Sk].
SUBADDITM VECrOR PROCESSES 87
By inequality (2.14) we have
S. 1 N
Z IQ n v Z ~ Nk (2, 1 4. + WN) V Z.
So by applying any f in E' and letting n go to infinity we get for w e
+ af
S S < ~EI[ 1k
F(Z) .Q lim inf = v z < lim sup =vz
~ n n k
Letting k tend to infinity we have
lim A't v Z) = f(Z) outside 2f. n
Next in view of the identity S,,In v Z + SnIn A Z = S.In + Z and (2.13) we also have for w C1 2f that
(2.16) A Z
n AZ).
For every to E 2, (S.1n)(W) A Z(£o) belongs to the weakly compact order interval [0, Z(o)]; hence there exists a subsequence (k,,) such that weak limit S,~.(w)lk,, A Z(w) exists. Lot Z'(w) be such limit.
For every f EE E', we have
lim~ Sk(w) A Z(w)) = f(Z'(£o)) = AzM) if W cl Of'
k,,
so Z' is weakly measurable. Since E is separable, Z' is also strongly measurable. Moreover, we have Z'(w) ~ Z(w) for each w E 2, so it follows that Z' is Bochner integrable.
Now for any fixed f GE E' we have for every A E 6Y that lim fAf(S,, / n) A Z)
fAf(Z') = fAf(Z). Since this last equality is valid for all f E E' we also have fA Z' fA Z and the validity of this equality for all A EE &Y imp~ies in turn that Z = Z' a.s. We have thus proved that (Sjn) A Z converges weakly a.s. to Z. By the first part of Lemina 1 we consequently have that (Sjn) A Z converges strongly a.s. to Z. To apply this to the strong convergence of (Sjn) we note
,,In _ Z11 < 11 L(INJZ, + W S,, 11 + 11Z 1
N) _ (IN IZ,. + W
n Nk n Rk N)11
(IN IZ, + W _ 1
N) *1 A Z11 + 11Z _ L(yN IZ,' + W
Nk ` n Nk ` N)11
where the last inequality follows from (2.14) and the fact that Nk ,Q n. By the equality (2.15) we obtain
lim SUP11 sil Z11 < 11E, [ Sk Z11 + 11z E, [ Sk
n k k
We already noted in (2.11) that EI[$k1k] converges strongly a.s. to Z and this fact completes the proof.
88 N. GHOUSSOUB A" J. MIC~ STEELE
As a corollary to Theorem 1 we note that the assumption of positivity can be weakened.
COROLLARY 1. If E has an order continuous norm and 8 is a Bochner integrable function such that for all n > 1
E~ 18 0i < S
3 0 n
then Sj n converges ~ with probability one.
PROOF. The corollary is immediate from the Mourier theorem and the fact that 1 S,' = S, In,_ .8 . 0' is a positive subadditive process.
The next theorem is the vector valued analogue of Kingman's ergodic decomposition of a real subadditive process.
THiBoREm 2. If E is weakly sequentially complete and (S,,) is a subadditive process such that
(2.17) sup 1 EIJIT3 I(Si Si 1 . 0)11 < 00
m
then there is a Bochner integrable random variable 8 such that for n > 0 we have
S,, > E! 118 . 0' and E(S) inf 1 E(S.).
m
PROOF. We let S,' 'S 0' S and note that by the subadditivity of S.
0 1 n
that S,,' is positive and superadditive. Also, since Si' Si'_ 1 0 0 = Si 1 0 Si +
S, we have
lllmi l(si, S,'~, 0 0)11 < 1127 I(Si 1 0 SM1 + mlisill so (2.17) yields
(2.18) sup 1 E111%I(Si' S,' , 0)11 < 00.
m
We will use Sn' to construct a S' such that
S,,' < 17 JS' . 0' for all n > 0
s 0
and such that E(S) = sup (1 / n) E(S,,'). By the correspondence between S, and S,,' the 8 required by the theorem is then given by 8 S, V.
We first note by (2.18) that
sup 1 1IE(S,)11 = sup Si' , 0)]11
n n
< sup 1 E1III I(Si' Si' , 0)11 < 00
n 9
so Proposition 2 implies that the superadditivity of E(S,') entails the convergence of
(11n)E(S,,') to sup(lIn)E(S.).
SUBADDITIVE VECTOR PROCESSES 89
As before we may assume without loss that E is separable. Consequently, E contains a quasiinterior point u (Schaefer [201 page 97). We will use this quasiinterior point to push through a truncation argument parallel to one used in the real case in Ackoglu Sucheston [11.
We let 0 = (1 / m)l ', 1 (Si' S,' 1 . 0) and note that the same computation as in the real case of Kingman's lemma will give
2:1 14, ' Oi > 1 _ n 1 'for 1 Q n < m.
(2.19) d_o m m ) S,,
Now for j > 0 the sequence (0.. Aju). is in the weakly compact interval [0,jul of Ll[E1 so by diagonalization procedure there are Xj C: Ll[E1 which are weak limits in V[E] of Aju) for a fixed subsequence (mk) and all j. By the weak lower semicontinuity of the norm in Ll[E] we have
(2.20) lini inf..E JJ0,., Ajull > E
We also have
sup,E[Ii~.11] .~ sup.,,Eiio.il < oo
thus the (~) are increasing and norm bounded so by the weak sequential completeness of E the (~) converge to their supremum V.
By (2.19) we have
10 .. k 0 01) A ju
Aju) . 0' 0) Aju) > (11
0 0
1 n 1 Aju
> ( Mk ) Sn
so taking weak limits with mk > oc we have
J:1 1x
8 0 j 0i > S,; Aju.
Now let j + oo to obtain the key relation
Si~181 . 01 > S,,for all n > 0.
8 0
Taking expectations further yields
ES' > sup L E[ Sn] n
and to get the opposite inequality it is enough to notice that
E[S] = sup E[~] r% sup E[O sup 1 E[ S.]
n
so the proof is complete.
90 N. GHOUSSOUB AND J. MIC~ STEELE
COROLLARY 2. For a Banach lattice E the following are equivalent:
(2.21) E is weakly sequentially complete.
(2.22) For every E valued subadditive Process (Sn) such that
SUp.11nilln_l(Si Si_1 . 0)11
i < oc we have
S.In converges in norm a.e.
PROOF. That (2.21) implies (2.22) follows from the theorem above and Corollary 1. For the converse it suffices to invoke Proposition 2.
For a real subadditive process the finiteness of the time constant inf(I1n)E(Sn) is sufficient for the a.s. convergence of SnIn. The next result shows that the spaces for which a corresponding result holds are precisely those isomorphic to L'[ g] for some tt, (i.e., A L space).
THE~ 3. For a Banach lattice E, the following are equivalent:
(2.23) E is isomorphic (as a Banach lattice) to an A L space.
(2.24) For every E valued subadditive process (Sj such that
inf(I1n)E(S.) exists, we have S,,In converges in norm a.s.
PROOF. By the same correspondence used in Theorem 2 we can work with positive superadditive processes (S,,) such that supj11n)E[Sj exists. That (2.24) follows from (2.23) is a consequence of the fact that an A L space is weakly sequentially complete and that there exists anf CE E' such that 11x11 C. f(ixi) for all x e E. To use this second fact, we note
sup 1 EIIY ni I(Si Si 1 o 0)11 IQ sup. 1 E17 Asi Si 1 0)
n n
sup. 1 E(f(SJ) sup 1 E(S.) < 00.
n n
To show the opposite implication we use the characterization of A L which says a space is A L if and only if every positive summable sequence is absolutely summable (see Schaefer [21] page 242). To that end, suppose there is a sequence (x,,) in E, such that 1,,,,,ox~ = x and 2:,,>oiix,,11 = 00. We can select an increasing sequence of integers (%) such that I',nt+.' ,11x,11 > 1 and by multiplying by coefficients less than 1, we may assume the sum equals 1.
For every k > 1 divide the interval (0, 11 into Mkll % subintervals; (A i, k)in k+.., such that the length of Ai,k 'S 11xJj. Now we can define an E valued process (Sj by
k 3 0%+1 X
(2.25) S,, n2:' 15Tk . 1141
If we take 0 to be the identity transformation on [0, 1], one can easily check that Sn
SUBADDITIVE VECTOR PROCESSES 91
is a superadditive process. Now
1 E(S11) = 1"~,E2:'i k+4,1
n k 1141 'A,, 7 "k ll*T, kmk+IXi < X.
On the other hand for any w EE [0, 1 ] we have for each k an ik such that w EE A 41 k~ and, consequently,
(2.26) 11 n S.MI1 = 1121 1 11Xt,'11
Since each of the surnmands is positive and of norm 1, the expression given in (2.26) tends to infinity with n, and the proof of Theorem 3 is complete.
3. Examples and applications. The usefulness of subadditive ergodic theory is substantially extended by a reformulation in terms of stationary pro6esses. After sketching this reformulation we will indicate several historically fecund sources of subadditive processes and analyze a problem from the theory of empirical distributions.
We will call a doubly indexed family of Bochner integrable random variables (Y,) s, t E= Z + a subadditive process provided
(3.1) Y,, ,Q Y,, + Y, whenever s < t < u
(3.2) (Y,,) has the same joint distributions as ( Y, ,,+ J.
To unite this definition with the one given in Section 1 we note first that (Y,,) can be extended to a process which satisfies (3. 1) and (3.2) for all s, t EE Z not just Z,. Next 2 can be taken to be the set of functions
x: {(s, t); s, t E= Z, s < t} > E
which satisfy
X(S, U) Q X(S, t) + X(t, U), (S < t < U).
The measure P on 2 is the measure induced by the process satisfying (3.1) and (3.2), and 0 is defined as the shift
(OX)(s' t) = X(S + 1, t + 1).
This correspondence is just the E analogue of the one introduced by Kingman, and for further details one can consult Kingman [18] pages 181 and 186.
For unity of this section on applications we restate our main result in terms of our second definition of subadditive processes.
THEoREm 1. For a Banach lattice E with order continuous norm, and for every positive process (Y,,) satis
_fying (3.1) and (3.2) we have Y% In converges in norm with ~ifity one.
This reformulation makes available a number of recipes for producing subadditive processes, some of which we now consider.
92 N. GHOUSSOUB AND J. MICHAEL STEELE
Unconstrained maximization. This is perhaps the most natural source of subadditive processes, and the theory of random series (as studied in Kahane [151) suggests many specific examples. To take one, suppose Xi i = 1, 2,. . . are Li.d. Bochner integrable random variables valued in a Banach lattice E. We let
y, MaXg :t116s+1Xs+1 + es+2X3+2 + + ex,
where the maximum is taken in the sense of the lattice E and extends over all choices of the signs c,. Even in the simple case E = L2[0, 21r] and X, = cos(x + Z) where Z, are i.i.d. rv's, it is useful to be able to conclude from Theorem 1 that n 'YO,, converges strongly with probability one.
Constrained minimization. This is a source of subadditive processes which goes back to the pioneering work of Hammersley and Welsh [14]. Suppose for each edge e of a connected graph G we associate a positive integrable vector valued random variable X(e). Given vertices v, v' of G we define
U(v, v) = infpY,..pX(e)
where the infinium is taken over all paths P from v to v' in G and the'sum is over all edges in P. When G is Z X Z with edge set given by joining a point to its four neighbors, we can then define a subadditive process by
Y.' WS, 0), (t, 0D.
A particularly interesting choice of X(e) is given by taking independent realizations of the absolute value of the Brownian bridge and where E = L1[0, 11. In connection with this specific choice it would be interesting to determine the value of the time constant
lim,'.CIE(YO,')
but this problem remains open.
Subadditive operators on random walks. Suppose (X) are Li.d. positive E,valued random variables and that M is a positive subadditive operator from E, to a Banach lattice E2. The process
y, ' = MX,.1 + X,+2 + +X)
will then be a subadditive process in E2. The operator M required above can be taken to be the Hardy Littlewood maximal operator on L` or any of the similar operators described by Stein [24]. Although the continuity of M will often allow one to side step Theorem 1, the use of subadditive processes retains a conceptual value and may prove valuable in more general contexts.
Beyond general sources of subadditive processes sketched above one can gain repeated inspiration from the examples given in Hammersley and Welsh [13], Hammersley [ 12], and Kingman [ 16, 17, 181.
We now turn to a more detailed discussion of a problem from the theory of empirical distributions which can be solved by means of vector valued subadditive processes and the application of Theorem 1.
SUBADDITIVE VECTOR PROCESSES 93
Uniformity classes and empirical distributions. We now assume (X,) is a sequence of i.i.d. random variables valued in R d with distributionju. A well known and much studied problem is the following:
Under which conditions on a class e of Borel subsets of R d does one have
SUPAIC1 1 2:"1 1 'A (X) g(A) 0 a.s. as n * oo? n
Many particular classes C, have been studied and a number of these are invariant under the group G of rigid motions of R ~. For example, the class of convex sets are studied in Rao [201, the half spaces in Wolfowitz [27], or many of the general classes studied in Vapnik and Chervonenkis [261, Steele [22], Dudley [91 or Dudley and Kuelbs [101.
There are also several classes of interest which are not invariant under G. Here one should consider the class of rectangles, class of lower layers [4, 23] or any class containing only a countable number of elements. Now for any class (2 we can define a new class eg = (B: B = gA, A EE 0_} by translating the elements of e. Similarly we have the translated discrepancy
0J9; W) SUPAECl 1 2:ni I'A(Xi) tt(A)I.
n
A class e for which Oje; w) 0 a.s. for the identity e is called a uniformity class, and the class formed by the union of all translates of 12 is called the closure of e under rigid motions (i.e., ~ = Ug,Geg).
It is natural to ask if ~ is a uniformity class provided e is a uniformity class. Unfortunately one can construct oven a singleton class e (A} such that 2 is not a uniformity class. (To prove this one can use the construction given in Steele [241 for a related problem.)
Even though ~ need not be a uniformity class, the natural measure on G (right Haar measure) shows that 2 shares some properties with 12.
d
THEomm 4. If (3 is a class of Borel subsets of the unit ball of R ' and the functions
0,,(g; LO) = SUPA.0,1 1 7,.' IIA(X) tt(A)1
n J_
are measurable then they converge in L '(G) with probability one.
PROOF. We first consider the process
y, SUPA.eil'i ."llg(A)(X,)
To check that Y,, is valued in Ll(G) we note
0 <.. Y1,1 I. ili ,+,1g(A)(Xi) + (t s)ja(g(A)).
94 N. GHOUSSOUB AND J. MICHAEL STEELE
Since 1 g(,)(x) is 0 except on a compact subset of G we have E~_,+11
V(G). For the second term we note, g(BA) E E
fGg(g(A)) dm(g) = fGfR'lg(A)(x) dtt(x) dm(g)
= fR"[fG'g(A)(X) dMW] dtt(x) = fR'[fG'g(A)(0) dMW] dWX) < 00 since g is a probability measure and since the last integrand does not depend on x. The remaining properties which show Y,, is an Ll(G) valued, positive, subadditive process are also easily checked. Hence by Theorem 1, we see 0Jg) = (I In) YO,,, converges strongly in V(G) with probability one (i.e., there is a 0 E= Ll(L1ffl)) such that f 10,,(g, w) .0(g, w)i dm(g) , 0 for a.e. W).
PROPOSITION 3. If in addition to the hypotheses of Theorem 4 we assume 03g is a uniformity class for each g 6 G then 0,, converges strongly to 0 with probability one.
PROOF. Let 0 be the limit of the 0,, guaranteed by Theorem 4. For any fixed g we have a.s. that
0(g, (0) = lim.~.0.(g, W) = 0, hence
0 = f f ago(g, w) dP dm(g) = fs] [ f co(g, w) dm(g) ] 0.
Since 4)(g, w) is positive this identity says 0(g, to) is the zero element of Ll(G) for a.e. w.
Theorem 4 and Proposition 3 have easy extensions to LP(G) for 1 IQ p < oo which we omit. As we commented before, there, is a singleton class e = {A) for which 0,, does not converge to 0 in V?(G), even though we now see it must converge to 0 in LP(G) for all 1 Q p < oo. This example is particularly useful to keep in mind considering the assumption of Theorem 1 on E that E have an order continuous norm (equivalently, 1 00 " E).
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SUBADDITIVE VECTOR PROCESSES 95
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DePARTmENT OF MAT~Tics DEPART~ OF STATisTics
UNIvERsiTY OF BRITISH COLUMBIA STANFORD UNIVERSITY
VANCOUVER, B.C. STANFORDt CALIFORNIA
CANADA